Turns against theory

Among the usual advantages of such expressions as Eq. (7-80) and (7-81), one is salient they show forth the invariance of p and w with respect to the choice of the basis functions, u, in terms of which p, a, and P are expressed. The trace, as will be recalled, is invariant against unitary transformations, and the passage from one basis to another is performed by such transformations. The trace is also indifferent to an exchange of the two matrix factors, which is convenient in calculations. Finally, the statistical matrix lends itself to a certain generalization of states from pure cases to mixtures, required in quantum statistics and the theory of measurements we turn to this question in Section 7.9. 

The process of total reflection of an incident wave in an optically dense medium against the interface of an optically less dense medium turns out to be of particular and renewed interest with respect to the concepts of nontransverse and longitudinal waves. In certain cases this leads to questions not being fully understood in terms of classical electromagnetic field theory [26]. Two crucial problems that arise at a vacuum interface can be specified as follows ... 

K.R. Popper, on p. 287 of Conjectures and Refutations (Harper and Row, New York 1968), argues that the least probable theories are the most valuable. He means that the more precise predictions a theory makes, the less one would bet on it a priori, but the greater its value when it turns out to be true, i.e., when its a posteriori probability after checking against reality is close to unity. 

Every chemistry student is familiar with the ideal gas equation PV = nRT. It turns out that this equation is a logical consequence of some basic assumptions about the nature of gases. These simple assumptions are the basis of the kinetic theory of gases, which shows that the collisions of individual molecules against the walls of a container creates pressure. This theory has been spectacularly successful in predicting the macroscopic properties of gases, yet it really uses little more than Newton s laws and the statistical properties discussed in the preceding chapters. 

The absolute values of Da for the phase separation and phase dissolution are plotted against e in log-log scale together in Figure 13. In the present case, the value of v turns out to be unity for the phase separation as well as for the phase dissolution, thereby confirming the prediction of linearized theory. A similar observation was also made by Snyder and coworkers (2Q) for the polystyrene (PS) polyvinyl methyl ether (PVME) blends. It is concluded that the diffusion rate is the same for the 60,000 and 100,000 molecular weight HPC specimens, if the diffusivities are referenced to the spinodal temperature. 

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